The different velocity transport schemes are demonstrated in this shallow-water model through two standard test cases.
Firstly, the second test from the suite of Williamson et al [1]}, which describes a zonal geostrophic flow.
This is a steady-state flow, so the evolved \(u\)  and \(h\)  fields can be compared with their initial values to compute errors due to the discretisation.
For full details of the initial conditions, see [1]}.
In Figure REF , the errors in \(\mathbf {u}\)  and \(h\)  are plotted after 5 days of simulation, for both the new schemes and the benchmark scheme.
The errors are computed at different spatial resolutions to approximate the order of accuracy of the overall model.
As in Section REF , the test was performed with a cubed-sphere mesh.
For all simulations we took the same time step of \(\Delta t=240\)  s.
Figure REF , shows the clear benefits of the two new schemes over the benchmark, improving the order of accuracy of the model from roughly first-order to approximately second-order for both schemes and both variables.
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